irish flute design optimization a major qualifying project submitted
TRANSCRIPT
Irish Flute Design Optimization
A Major Qualifying Project submitted to the faculty
of Worcester Polytechnic Institute in partial fulfillment of the
requirements for the Degree of Bachelor of Science.
Submitted By:
Ying Bao
Kiara Gravel
Ian Jutras
David Knight
Submitted To:
Advisors: Professors John Sullivan and David Planchard
April 23, 2013
i
Abstract
The goal of the project was to improve the intonation and ergonomics of an Irish flute as
well as assess the feasibility of rapid prototyping a flute. SolidWorks was used to model existing
Delrin and PVC flutes. A rapid prototyping machine was used to create a physical model of the
six holed Pratten-Style Delrin polymer flute based on the SolidWorks data. Using the rapid-
prototyped Irish flute and PVC flute, the group collected frequency-based data. Frequencies
were collected in the first and second octaves and compared to the A440 Hz standard of tuning
musical instruments. Through studying acoustic principles of an open tube, the group
developed equations to predict the distance from each of the tone holes to the embouchure
hole based on standard frequencies as well as various dimensions within an Irish flute.
Parameters of the flute such as the bore diameter, tone hole diameter, cork position, chimney
height, and length of the flute were isolated and analyzed. These equations were then linked
through a design table in SolidWorks to the 3D model and used to design and create a new,
optimized, rapid prototyped flute. The design table facilitated the process of making
dimensional modifications to the flute without having to manually calculate the values and
input them into a model. The final result was an optimized Irish flute that required minimal
tuning. The group gained a deep understanding of the relationships that exist between
parameters of the Irish flute and the acoustics through the instrument. Recommendations for
further studies on the Irish flute were included.
ii
Executive Summary
The project goal was to optimize the existing Irish flute from the engineering
perspective. This was accomplished by developing and applying mathematical modeling as well
as modern manufacturing technology, such as 3D rapid prototyping. The team began the
project by studying the fundamentals of flute acoustics together with the factors that could
affect overall flute performance. These factors included materials, embouchure hole shape and
placement, bore shape and diameter, tone hole placement, tone hole size, and chimney
heights.
Six Irish flutes were used in the project. The Delrin keyless flute, was modeled in
SolidWorks 2012. To create the initial rapid prototype, this SolidWorks model was printed on
an Objet260 Connex Rapid Prototype machine. The team used 220-grit sandpaper to smooth
out the surface and remove any remaining support material. Two types of flute playing devices
were designed for the flute testing in order to ensure the consistency of the airflow.
Existing mathematical models failed to match actual measured values within reasonable
error. Consequently, the team developed its own mathematical models. Equations predicting a
flute’s dimensional parameters based on desired frequency were successfully developed. To
facilitate implementation of these equations into a solid model of the flute, a user-friendly
SolidWorks Design Table was created. Using the table, flute dimensions could be easily
changed and the corresponding locations of tone holes adjusted automatically to the changes.
Design tables like this can allow the manufacturer to input information about the player, such
as finger spacing, which can allow well-intonated flutes to be custom-designed and rapid-
prototyped for individual players.
The second prototype was designed using the SolidWorks design table. This prototyped
flute was easily playable across the entire note range while producing a clear tone. A tuning
slide with a fully-extended length of 30 mm was implemented to ensure that the flute could be
tuned to concert pitch regardless of temperature. The flute was designed so that under
standard temperature, the slide would be extended to 10 mm, which allowed the flute’s
frequency to be raised as well as lowered. The second prototype also featured an extended
chimney height on the tone hole farthest from the embouchure hole. In order to raise the
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volume of the E-note played by the last tone hole, the hole diameter was increased. The raised
chimney height was implemented to counter the raised frequency of the E-note which would
have resulted from the increased E-hole diameter.
Due to limited time, the team was not able to acquire sufficient testing data to develop
a complete mathematical model for chimney height prediction. Future research could focus on
creating this model, as well as the modification of the embouchure hole, bore profile, and bore
roughness.
iv
Acknowledgements
This project would not have been possible without help and funding from the ME
department. In addition, special thanks to Erika Stults, who supervised the rapid prototyping
process and performed extensive cleaning. Peter Hefti donated time and materials to the
group by assembling the regulator for the flute playing device. Windward Flutes loaned the
group two Irish flutes for testing. Jeffrey Peters acted as a test player. Finally, our thanks go out
to our advisors, Professors John Sullivan and David Planchard, who provided invaluable insight
and guidance.
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Table of Contents
Abstract ............................................................................................................................................ i Executive Summary ..........................................................................................................................ii Acknowledgements......................................................................................................................... iv
List of Figures ................................................................................................................................. vii List of Tables ................................................................................................................................... ix
Nomenclature .................................................................................................................................. x
Introduction .................................................................................................................................... 1
Background ..................................................................................................................................... 3
History ......................................................................................................................................... 3
Acoustics ..................................................................................................................................... 3
Material ....................................................................................................................................... 5
Embouchure ................................................................................................................................ 6
Bore ............................................................................................................................................. 9
Tone Hole, Placement, Size and Chimney Heights. .................................................................. 12
Methods and Procedure ............................................................................................................... 14
Irish Flutes Used ........................................................................................................................ 14
Initial Rapid Prototyping ........................................................................................................... 16
Design and Fabrication of the Flute Playing Device ................................................................. 18
Testing Previously Developed Equation Models ...................................................................... 21
Initial Mathematical Modeling.................................................................................................. 21
Frequency Gathering of PVC, Delrin®, RP, Olive, and Mopane Material Flutes........................ 27
Human Player Frequency Testing ............................................................................................. 27
Flute Playing Device .................................................................................................................. 28
Chimney Height ......................................................................................................................... 28
Bore Extension Testing ............................................................................................................. 29
Temperature ............................................................................................................................. 30
Cork Displacement .................................................................................................................... 30
Human Player Averaging ........................................................................................................... 30
Final Rapid Prototyping ............................................................................................................. 30
Results ........................................................................................................................................... 33
Results of Initial Prototype ....................................................................................................... 33
Previously Developed Equations .............................................................................................. 34
Uncertainty Calculations ........................................................................................................... 35
General Frequency .................................................................................................................... 36
vi
Cork Displacement .................................................................................................................... 38
Bore Extension .......................................................................................................................... 40
Temperature ............................................................................................................................. 44
Human Player Averaging ........................................................................................................... 45
Chimney Height Results ............................................................................................................ 46
Predicting Frequency ............................................................................................................ 46
Predicting Chimney Height ................................................................................................... 47
Flow Inside the Bore ................................................................................................................. 48
Implementation of the Equations for Manufacturing .............................................................. 49
Results of Final Prototype ......................................................................................................... 51
Conclusions ................................................................................................................................... 54
Final Prototype Assessment ...................................................................................................... 54
Manufacturing Feasibility ......................................................................................................... 54
Recommendations and Future Work............................................................................................ 55
Chimney Height Prediction ....................................................................................................... 55
Embouchure Hole ..................................................................................................................... 55
Bore Profile Modification .......................................................................................................... 56
Future Work on Bore Extension ................................................................................................ 58
Future Work Based on Final Prototype .................................................................................... 58
Work Cited .................................................................................................................................... 59
Appendix A .................................................................................................................................... 61
vii
List of Figures
Figure 1: Nomenclature Visualization ............................................................................................. xi Figure 2: Basic Parts of the Flute .................................................................................................... 1
Figure 3 Notes and Fingering Chart for an Irish D Flute ................................................................. 2
Figure 4: Sine Wave ........................................................................................................................ 3
Figure 5: Idealized Model of a Standing Wave in a Simplified Flute .............................................. 4
Figure 6: First Three Modes of Vibration ........................................................................................ 5
Figure 7: Two Semicircles embouchure hole (McGee) ................................................................... 7
Figure 8: Rounded Rectangle embouchure hole (McGee) ............................................................. 7
Figure 9: 19th Century Flute (Transverse Flute in F) ....................................................................... 8
Figure 10: Cylindrical vs. Conical Bore .......................................................................................... 10
Figure 11: Pressure Waves through Cylindrical and Conical Bores (Ayers) .................................. 11
Figure 12: PVC D Flute Manufactured by repurposeeverything.org ............................................ 14
Figure 13: Delrin D Flute Manufactured by David Copley in Ohio................................................ 14
Figure 14: African Blackwood D Flute Manufactured by Bryan Byrne in Vermont ...................... 14
Figure 15: Mopane Wood D Flute Manufactured by Windward Flutes ....................................... 14
Figure 16: Olive Wood D Flute Manufactured by Windward Flutes ............................................ 15
Figure 17: Olive Wood D Flute Modified to Play in the Key of Eb ................................................ 15
Figure 18: Method of Determining Angle of Undercut ................................................................ 16
Figure 19: SolidWorks Models of Delrin® (top) and Initial Rapid Prototype (bottom) ................ 17
Figure 20: Copper Pipe Constant Playing Apparatus .................................................................... 19
Figure 21: Completed Flute Playing Device on a PVC Flute .......................................................... 20
Figure 22: Correction Length used in Determining Tone Hole Location ...................................... 22
Figure 23: Change in Wavelength Node due to Reduction in Diameter ...................................... 23
Figure 24: gString Tuner Application ............................................................................................ 27
Figure 25: Set up using Cardboard to Increase Chimney Heights ................................................ 29
Figure 26: Final Rapid Prototype Cross Section with Tone Hole Locations .................................. 31
Figure 27: Close of End Joint in Final Flute ................................................................................... 32
Figure 28: Magnification of Layers Created by Rapid Prototyping Vertically ............................... 33
Figure 29: Difference in Frequency vs. Actual Frequency for the Mopane Wood D Flute........... 37
Figure 30: Difference in Frequency vs. Actual Frequency for the Olive Wood D Flute ................ 37
Figure 31: Difference in Frequency vs. Actual Frequency for the Olive Wood Eb Flute .............. 38
Figure 32: Change in Frequency Due to Cork Position ................................................................. 40
Figure 33: Change in Frequency over the Base Frequency (Fbt) with Respect to the Base Frequency (Fbt) for a Bore Extension of 3.82 mm ......................................................................... 42
Figure 34: Change in Frequency over the Base Frequency (Fbt) with Respect to the Base Frequency (Fbt) for a Bore Extension of 7.03 mm ......................................................................... 42
Figure 35: Change in Frequency over the Base Frequency (Fbt) with Respect to the Base Frequency (Fbt) for a Bore Extension of 10.52 mm ....................................................................... 43
Figure 36: Temperature Variations within the Bore of the Flute while being Played .................. 44
Figure 37: Difference between Played Frequency and the Actual Frequency in Average Human Player Test on the Initial Rapid Prototyped Flute ......................................................................... 45
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Figure 38: Difference between Played Frequency and the Actual Frequency in Average Human Player Test on the Delrin® Flute ................................................................................................... 46
Figure 39: 3-D graph showing relationship among distance, chimney height and frequency ..... 47
Figure 40: Frequency Change due to Chimney Height Increase in the E4 Tone Hole .................. 48
Figure 41: Calculations Inserted into Design Table ...................................................................... 50
Figure 42: Final Rapid Prototype................................................................................................... 51
Figure 43: Magnification of Layers Created by Rapid Prototyping Horizontally .......................... 52
Figure 44: Frequency Data for the Final Rapid Prototype ............................................................ 53
Figure 45: Bore Profiles with Varying Tapers (Hopkin) ................................................................. 57
Figure 46: All required values and equations needed to compute tone hole placement in an Excel spreadsheet ......................................................................................................................... 61
Figure 47: Mopane Wood D Flute Frequency Test ....................................................................... 62
Figure 48: Olive Wood D Flute Frequency Test ............................................................................ 62
Figure 49: Olive Wood Eb Flute Frequency Test ........................................................................... 63
Figure 50: Difference between Predicted Change and Actual Change in Frequency due to Bore Extension ....................................................................................................................................... 63
Figure 51: Frequency Data Collected for Chimney Height Increases ........................................... 63
ix
List of Tables
Table 1: Testing of Forster's Effective Length of Tone Hole Equations ........................................ 34
Table 2: Calculation of Distance from Embouchure Hole to the Left of the Cork ........................ 35
Table 3: Relative Uncertainty in Tone Hole Placement Calculations............................................ 36
Table 4: Frequency Played Relative to Cork Position ................................................................... 39
Table 5: Change in Frequency with Respect to Cork Position ...................................................... 39
Table 6: Results from Bore Extension Test ................................................................................... 41
Table 7: Difference in Frequency from Base................................................................................. 41
x
Nomenclature
Abe (mm2) = Area of the bore at the embouchure hole
Ae (mm2) = Area of the embouchure hole
Aest (mm2) = Estimated area of the bore at the active tone hole using the value dest
Ah (mm2) = Area of the active tone hole
dbe (mm) = Diameter of the bore at the embouchure hole
de (mm) = Diameter of embouchure hole
dend (mm) = Diameter of the end of the bore
dest (mm) = Estimated diameter of the active tone hole
dh (mm) = Diameter of the active tone hole
Ex (mm) = Extension distance of the tuning slide
Fb (Hz) = Base frequency of the flute: frequency played when all tone holes are closed
Fbt (Hz) = In tune base frequency corresponding to each tone hole
Fd (Hz) = Desired frequency for the active tone hole
hc (mm) = Height of the chimney above the outer bore
ho (mm) = Wall thickness at embouchure
hend (mm) = Wall thickness at end of flute
hest (mm) = Estimated wall thickness at the active tone hole
Δle (mm) = Length correction for the embouchure
Δlh (mm) = Length correction for the active tone hole
Δlhemb (mm) = Length correction for the end of the flute with all tone holes closed
La (mm) = Length between embouchure hole and active tone hole
Ls (mm) = Length of the actual wavelength
c (m/s2) = Speed of sound
xi
Figure 1: Nomenclature Visualization
1
Introduction
The Irish flute is a woodwind instrument that is traditionally used to play Irish music.
The flute consists of an embouchure hole where a player’s lips are placed and six tone holes
which can be covered by the player to produce different musical notes. The Irish flute is similar
to an open tube; one end remains open while the other “open” end is at the embouchure hole.
The basic parts of the flute are shown in Figure 2.
Figure 2: Basic Parts of the Flute
The design of the Irish flute had potential for intonation improvement. While keeping
the Irish flute within its true definition as a keyless instrument, the group aimed to optimize the
flute’s intonation. Intonation is the relation in pitch of tones to their accepted standard values.
The group decided to alter the different aspects of the design to improve the intonation with
respect to accepted pitch frequencies based upon the A440 Hz standard.
The basic notes that were tested were D, E, F#, G, A, B, and C#. Each of these notes can
be produced by covering the tone holes and blowing over the embouchure hole. Figure 3
illustrates the tone hole combinations used to produce each note. The dark circles represent
tone holes that are covered while the white circles with a black outline represent uncovered
tone holes.
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Figure 3 Notes and Fingering Chart for an Irish D Flute
By increasing the air pressure blown across and into the flute, the second octave for
each of these notes can be produced. Various dimensional parameters were isolated for the
first and second octaves of the flute, and the effects of their modifications were studied.
Parameters studied included changing the cork location and the length of the tube. In addition,
the embouchure hole and tone holes were tested by changing their profile and location. The
chimney height, which is the thickness of the flute at the tone hole, for each note was increased
to study the frequency effects based on the thickness.
To begin the study, a flute was modeled in SolidWorks. The model was based on a six
holed David Copley Pratten-Style Delrin® polymer flute. A rapid prototype (RP) was created
from this model. Frequency data were collected through testing of the prototype, and
mathematical relationships between the parameters were identified. Relationships were also
analyzed for the Delrin® and PVC flutes. The final RP was created using the identified
mathematical relationships. This flute was designed to be played in tune with minimal
embouchure adjustment or tuning, and to fit the player’s hand more comfortably.
3
Background
History
The design of the Irish flute is reminiscent of early flutes from the Renaissance Period
(14th to 17th century). During that time period, flutes were typically used for chamber music
and military bands. Since the bore was cylindrical, the upper and lower octaves were
frequently not in tune (McGee). To improve the tuning, these Irish flutes were replaced by the
“Böhm-style” keyed flutes in 1847. The Böhm flute became popular, and the prices of older-
style flutes drastically decreased. As a result, less affluent musicians began purchasing the
older-style flutes in large numbers. Poor economic conditions in Ireland caused a heavy
concentration of these flutes to be bought and used by Irish musicians. This economic situation
caused this simple flute style to be referred to as the Irish Flute. The modern Irish flute is
typically made of wood, metal, or heavy plastic (such as Delrin®). It has six tone holes and no
keys. While the bore is usually conical, less expensive flutes may have cylindrical bores (Xorys).
Acoustics
Sound is produced when vibrations travel through air. These vibrations can be
represented as a sine wave which can be seen in Figure 4 below.
Figure 4: Sine Wave
4
This wave has an amplitude “A” which correlates to volume of the sound. The period
“T” is length that one sine oscillation takes to complete a cycle. The inverse of the period is the
frequency of the wave. This frequency is what determines the actual note or sound that is
heard. A longer sine wave cycle denotes a lower frequency resulting in a lower-pitched sound.
Two notes that are an octave apart will differ in frequency by a factor of two.
In a flute the vibration of the air is caused by a traveling pressure wave created when
the player blows air into a flute. A rise in the wave will signify when the air molecules are
forced together (“compressions”) while the decrease in the wave will signify when the air
molecules are forced apart (“rarefactions”). During flute intonation optimization, the measured
frequencies of the notes of the flute must stay in tune relative to the standard tuning of A=440
Hz. In air columns such as what is found in the flute, a traveling wave will reflect back on itself
every time it reaches a boundary (Hopkin). Boundaries within the flute include the walls of the
tube, the cork located near the embouchure hole and the end of the flute open to atmospheric
pressure. This open end acts as a boundary because the air column is no longer constrained to
the inside of the flute. Despite the presence of the cork, the flute is analyzed as an open tube
on both ends. The embouchure hole acts as the open end adjacent to the cork. Steady-state
vibrations exist as result of traveling waves reflecting back and forth in the air column. To
create a standing waveform, interacting wave fronts reinforce or cancel each other within the
bore (Hopkin). An example of a standing wave can be seen in Figure 5. This standing wave
creates the fundamental pitch as well as overtones that are heard.
Figure 5: Idealized Model of a Standing Wave in a Simplified Flute
The different modes of vibration occur because multiple tones arise from reflecting
patterns. Fundamental tones and overtones occur together, but the fundamental tone is the
most audible. This fundamental tone creates the pitch that is heard. Several modes of
vibration are possible and each mode is a pattern of vibratory movement in the standing wave.
The simplest and most prominent mode is the first mode, which is responsible for the pitch.
5
The second mode is an octave above the fundamental. The third mode of vibration occurs at
the next-highest overtone, which is a 12th, or 12 musical steps, above the fundamental. These
three modes of vibration can be seen in Figure 6.
Figure 6: First Three Modes of Vibration
In standing wave patterns, there are points of maximum movement called antinodes, as well as
points of minimum or no movement called nodes. Longitudinal modes are the most important
in air columns. For an air column to vibrate freely in a particular mode of vibration, it is best for
the points located along the antinodes to not be disrupted (Hopkin). Pinpointing the areas
where nodes and antinodes occur can allow the researcher to alter the inside of flute and avoid
disturbing the waveform.
Material
The material of woodwind instruments presents a point of contention between
musicians and scientists. While scientists claim that material choice offers very little to
instrument tone, musicians swear that the difference is perceivable by both feel and sound.
In a 2001 study, Gregor Widholm matched seven identical flutes of different materials
with seven professional flautists. Each flautist played all seven flutes, and the sound samples
were analyzed by a panel of 15 professional flautists, including the seven test players
themselves. After analysis of flutes made of solid silver, plated silver, 9 karat gold, 14 karat
gold, 24 karat gold, solid platinum, and plated platinum, it was determined that while “sound
6
analysis pointed out big differences in the sound level and sound color of played tones caused
by the player,” the material caused tone color differences that were “just measurable but not
perceivable” (Linortner).
Other professional flute makers have their opinions on material versus player. Terry
McGee, an experienced and prolific flute maker from New South Wales, Australia, agrees with
the scientists, but with a small caveat. McGee states that “the performance of a flute is going
to be principally determined by its shape…providing it’s smooth, doesn’t leak, and it’s strong
enough not to vibrate and rob energy from the vibrating air column.” (McGee) On the other
hand, Malcolm Tattersall indicates in a 2007 article that while material may not affect tone, it
may affect the process by which the flute is made. For example, some materials yield a more
precisely manufactured result which in turn may affect tone. Furthermore, a player’s
“preconceptions and wishful thinking” about the instrument will have an effect on both the
quality of the playing and on the player’s perception of tonal quality (Tattersall).
In 1998, Cocchi and Tronchin performed a sound analysis of two flutes: one made of
“light alloy” and one made of silver. Using homemade software, the researchers found that
“the silver flute contains more high frequency and shorter transient than the light‐alloy flute”
(Cocchi, Tronchin). Through the study, material was found to have some effect on the
measured frequencies in the flute. Since the flute material appears to only slightly affect tone,
where it is minimally perceptible to the human ear and difficult to collect data with the group’s
measuring devices, the analysis for the flute was reduced to not focus on material selection as
an area of potential improvement.
Embouchure
According to the fifth edition of The Concise Oxford dictionary of music, “embouchure”
is defined as the mode of application of the lips, or their relation to the mouthpiece in a brass
and woodwind playing instrument (Kennedy). The embouchure hole is where energy is
imparted to the bore of the flute through blowing air. The energy is used to create musical
pitches. There are several aspects related to the embouchure that have great influence on the
tone production of a flute. These aspects can be divided into two major categories based on
whether or not the tone production involves “human factors”.
7
Without adjusting the shape of the embouchure hole, there are a number of player-
specific factors that influence the tone of the flute:
How high or low the lip is placed on the flute relative to the embouchure hole
The amount of the embouchure hole that is covered by the lip
Relation of the lips to the embouchure hole: positioning the embouchure hole
slightly to the left or right of the center of the lips
The angle of the lips on the flute’s bore
The width and the depth of the gap between the lips
Whether the flute is played with a “smiling,” “straight,” or “sad” embouchure
The angle of the air jet. This can have an impact on tone production, not only in
changing octaves, but also in the resonance of a note (Wilcocks, 2006).
Without any human factors, the design of the embouchure hole itself can have great
influences on the overall intonation of the flute. The classic elliptical embouchure found on
19th century flutes tends to be too noisy and unresponsive, even though it can produce an
attractive dark sound when played in the Irish style. The modern-designed embouchure hole is
a bit louder, much easier to play, and more responsive with faster articulation compared to the
traditional version.
Figure 7: Two Semicircles embouchure hole (McGee)
Figure 8: Rounded Rectangle embouchure hole (McGee)
8
Figure 7 and Figure 8 show two typical types of modern-cut embouchure holes in
modern Irish flutes. The “two semicircles” embouchure hole provides a useful increase in area
over the traditional elliptical hole, increased width of the “edge”, and better dissociation
between edge and sides. The edge is defined as the opposite side of the hole from the player’s
lips. The rounded rectangle embouchure hole provides a further increase in area, an even
wider "edge", and even better dissociation between edge and sides which increased the ease of
playing the flute and volume. When modern embouchure holes are compared with 19th
century embouchure holes, slight variations to the outer shape of the embouchure hole exist
with an increased area.
Figure 9: 19th Century Flute (Transverse Flute in F)
For example, the smaller embouchure hole dimensions make focusing of the air stream easier.
A deeper chimney produces more desirable tone. The undercutting compensates for the
smaller dimensions and deeper chimney. The undercut edge sharpens the edge angle. A more
rectangular hole maximizes the cross sectional area and widens the edge. The increased cross
sectional area improves the volume of the sound and the overall easiness of playing. In
addition, when the player's side of the hole is thinned and contoured to get lips closer to edge,
focusing becomes easier and lip support is increased. The rounded sides of the embouchure
hole, both inside and out, reduce wind noise (Mcgee).
9
The location of the embouchure hole relative to the cork also contributes to tone
production. Based on the book Fundamentals of Musical Acoustic, the distance from the
embouchure hole to the inside face of the cork, which is the theoretical starting point of the
sine curve for each note, is defined as the embouchure hole length correction Cemb. The
formula for calculating the distance is as follows:
e
eb
emb HWD
dC *)
*(
2
The embouchure hole dimensions are defined by a rectangular shape of width “W”,
breadth “D” the chimney height, “H”. The radius of the air column is defined by “dbe”. “He” is
the effective height of the chimney which includes the chimney height of the flute plus the
player’s lower lip thickness
The magnitude of Cemb varies with frequency. The most important function of the head
joint cork, along with the player’s lip position, can be adjusted to provide a suitable value for
Cemb which is necessary for better alignment of the nodes. Most flutes have a Cemb value of
around 50 mm (Benade).
Bore
The inside of the pipe, or “bore,” was another important consideration in the redesign
process. The bore is the route that much of the air takes when a person blows through the
embouchure hole. In reality, the bore could take any shape from cyclical to random twists and
cuts, but outlandish shapes are not normally used. Besides the difficulty in manufacturing,
these outlandish shapes are a problem because any sharp edges inside a tube can cause
disruptions to the sound waves traveling through the bore. Because a harmonic overtone
series is desired, most Irish flutes have been produced with cylindrical or conical bores with
some alterations in the taper at strategic points. Outlandish bore shapes are difficult to
manufacture (Hopkin).
Surprisingly, cylindrical and conical bores like the one seen below in Figure 10 exhibit
very similar behaviors.
10
Figure 10: Cylindrical vs. Conical Bore
When calculating the frequencies for both cylindrical and conical bores in their modes of
vibration, one may observe that both carry the same general equation:
L
nvnf
2)(
where “n” is the number of the mode being evaluated, “v” is the speed of sound at 25 °C (346
m/s), and “L” is the tube length. A true conical bore would also have a characteristic of a closed
end, but in wind instruments such as the Irish flute, the frequency is determined in the above
manner.
The cross-sectional area of the bore is far more important than its shape. As long as the
cross-sectional area of a cylindrical bore matches that of a square bore, there will be little
acoustic effects. For example, consider two uniformly-increasing or -decreasing bores, one
being square-shaped and the other being circular. As long as the cross sectional areas at
corresponding ends of the two bores are equal and these bores taper at the same rate, the
same tones will be produced (Hopkin).
Conical and cylindrical bores are the most efficient means of creating the fundamental
tones in a wind instrument with no keys. Slight differences exist between these two bore
profiles as the ability to manufacture different-shaped bores is difficult. These differences can
be attributed to the resonant modes or excitement in the overtones. The different bore shapes
11
have slightly different effects on the pressure waves that will travel through the bore. These
pressure variations are illustrated in Figure 11. The trend for resonant modes in an open tube is
a sine wave with the function f(x) =sin(x). For a true conical bore, one end is closed, and
pressure variations are depicted as following a pattern with the function f(x) = sin(x)/x.
Although the Irish flute was considered open at both ends, it is important to note the pressure
waves for the conical bore shown in the illustration below. If the closed end in the illustration
was no longer depicted, the resulting pressure variations would be very similar to those of a
conical bore flute open at both ends. The difference would be in the amplitudes while both
profiles would follow similar sine waves (Ayers).
Figure 11: Pressure Waves through Cylindrical and Conical Bores (Ayers)
While changes to the bore profile were design consideration, some argue that the bore
has no effect on the sound produced by the flute. The “ideal” flute tone, known as the “son
plein,” has a quality of tone resembling that of a clarinet. The tone can be more difficult to
achieve with a cylindrical bore than with a conical bore, yet it is not impossible (Welch). Welch
12
claims that the sound depends more on the skill of the flute player, and that the strength of the
lip and size of the holes are more important than the bore profile (Welch).
A conical bore can be thought-of as a cylindrical bore with a taper. A conical bore is
observed to be more desirable in flute design, as it seems to improve intonation (McGee). A
common design utilizes a cylindrical bore from the embouchure hole to the first finger hole,
where the bore becomes and remains conical but is still open at one end (Healy).
When looking at the diameter of the bore, Mark Shepard determined that an ideal
length to diameter ratio is 23:1. This is due in part to the fact that pipes with shorter lengths
and larger diameters will produce poor overtones, while a flute with a longer length and shorter
inside diameter will produce a clearer, higher volume note. However, a tube that is too long
and has too small a diameter will produce breaks in the harmonics. Most flutes alter the bore
at about 1/5 of the length from the embouchure hole with a taper that ultimately reduces the
diameter by 10%. By providing such a taper, the pitch of the lower tones can be lowered
without a significant difference in the higher tones (Hopkin).
Tone Hole, Placement, Size and Chimney Heights.
Even though flutes follow the same principles as open tubes they still exhibit some
different behaviors. The holes are located along the cylindrical face of the tube, which changes
the way the sound waves travel though the tube. The actual distance from the embouchure
hole to the next open tone hole is less than the distance that would correspond to a tube with
no holes that plays the same note. There is a theoretical end correction factor which must be
added to the actual distance in order to determine the note that will be played (Hopkin).
The tone holes that are covered by the player’s fingers determine the note that will be
played when air is passed over the embouchure hole. The location of the tone hole is the major
factor that determines the note that is heard. The closer the hole is to the embouchure hole,
the higher the frequency that will be heard.
Another way to vary the pitch is by varying the diameter of the tone hole. A hole with a
larger diameter will sound a higher pitch. If the diameter of the tone hole is equal to the
internal diameter of the flute at the tone hole, the sounded note will have a frequency equal to
the frequency played by a tube of a length equal to the distance between the embouchure and
13
tone holes (Hopkin). The closer the tone hole is to the internal diameter of the flute, the lower
the theoretical end correction factor. With a larger tone hole, a louder volume and a richer
tone are produced. However, changes in tone hole size can affect the comfort of the player.
The pitch can also be raised or lowered depending on the height of the material around
the hole. This wall thickness can also be referred to as the chimney height. In a typical flute
this will be equal to the wall thickness of the tube of the flute. However, if the chimney height
is reduced, it will raise the pitch of that note. The reverse is also true. If the height of the
material is increased around the hole, it will lower the pitch of the note (Hopkin).
14
Methods and Procedure
Irish Flutes Used
Throughout the project, six main keyless flutes were analyzed and used to record data.
One of the primary flutes used was a D flute made out of a ¾” diameter piece of PVC pipe. This
flute was purchased from repurposeeverything.org. Other flutes were also created by the
group using ¾” diameter PVC pipe. A Pratten-syle D flute made out of a Delrin® polymer was
also extensively used in testing. Additional flutes included D flutes made out of African
Blackwood, Mopane, and Olive wood. The Olive wood flute had separate segments and could
be modified to become an Eb flute. These flutes can be seen in Figure 12, Figure 13, Figure 14,
Figure 15, Figure 16, and Figure 17, respectively.
Figure 12: PVC D Flute Manufactured by repurposeeverything.org
Figure 13: Delrin D Flute Manufactured by David Copley in Ohio
Figure 14: African Blackwood D Flute Manufactured by Bryan Byrne in Vermont
Figure 15: Mopane Wood D Flute Manufactured by Windward Flutes
15
Figure 16: Olive Wood D Flute Manufactured by Windward Flutes
Figure 17: Olive Wood D Flute Modified to Play in the Key of Eb
Throughout the project, it was essential to create accurate 3D CAD models of the Irish
Flute. The group used SolidWorks 2012 to create these models of each flute. The models were
a convenient way to keep track of the dimensions of each flute. While the group measured
each of the flutes as accurately as possible, some dimensions were difficult to obtain.
Measuring the undercutting found at the embouchure hole of the Delrin® flute was one
challenge. Other small dimensions such as angles within the bore were measured to the
group’s best ability. A technique was developed in order to determine the angle of undercut
for the embouchure hole. This process used a rigid card to translate the angle of undercut to
an angle that could be measured on a flat surface. The angle α, seen in Figure 18, was
determined by measuring the distances “A” and “B” and using trigonometry to determine the
value of the angle.
16
Figure 18: Method of Determining Angle of Undercut
Initial Rapid Prototyping
An important goal of the project was to assess the feasibility of rapid prototyping an
Irish flute. A main concern was that the rapid prototype material would allow too much air to
escape through the joints of the flute. Another issue was whether the material would be too
fragile to be viable for flute production.
The advantage of rapid prototyping is that features such as extended chimney heights
on tone holes can be made very easily. Any type of variable bore profile can be used because
machining considerations do not need to be made. Using a rapid prototype machine also
allows flutes to be created with higher accuracy than traditional methods such as turning on a
lathe. Rapid prototyping can result in a more consistent and customizable design.
The group based its initial prototype design on a David Copley Delrin® flute. The CAD
model of this flute can be seen in Figure 19.
17
Figure 19: SolidWorks Models of Delrin® (top) and Initial Rapid Prototype (bottom)
18
The Delrin® flute seemed like a common design, yet it was more complex than the
simple flutes made out of PVC tubing. This design was modeled as precisely as possible before
it was rapid prototyped. In the model, a gap of 0.0254 mm was used between all the
overlapping joints. This was done to compensate for the tolerance allowance of the rapid
prototype machine. The flute model was divided into six smaller parts before it was printed.
This was done so that all the parts could be printed vertically, and so that it would be possible
to create replacement parts without having to reprint larger sections. Furthermore, when
printed vertically the prototype was symmetric about the axis of rotation. This is because the
layers created by the printer were perpendicular to the axis of rotation. It was feared that
layers parallel to the axis of rotation may cause grooving in the bore. The flute was printed
with a cork built into the flute at a fixed location.
The group’s rapid prototype was created using an Objet260 Connex Rapid Prototype
machine. The material used was called VeroWhite. The .stl files were saved with a tolerance of
0.1524 mm, which corresponded to the tolerance of the RP machine. The group chose to use
the finest possible tolerance in an attempt to reduce any effects of surface roughness on the
model.
There were a number of different finishing techniques that were applied to the rapid
prototype. The first technique was to use an automotive buffing compound to smooth out the
flute. This did not provide a significant amount of refinement. The next method was to use
220-grit sandpaper. This resulted in a much smoother finish than the printer was able to
produce. This finishing did not remove too much of the material on the flute, so the parts still
fit together fairly well, and the flute played as expected.
Design and Fabrication of the Flute Playing Device
In order to ensure consistent testing, a method to provide a steady note on the flute
was needed. The first attempt was to take ¼” copper pipe and bend it around the flute. An air
hose would then be connected to the end. The end that was above the embouchure hole was
bent slightly to angle the air into the embouchure hole. This can be seen in Figure 20.
19
Figure 20: Copper Pipe Constant Playing Apparatus
This apparatus functioned in the upper octave, but it did not produce very consistent
results and it could not play the lower octave. The range also tended to be higher than the
normal range of the flute.
The flute playing device was conceived out of the need for easily measurable and
repeatable flute frequencies. It was determined that in order to reproduce the human ability to
blow into the flute, the machine would need a pressure source, a perturbation chamber, a
nozzle, and a simulated lower lip. The upper lip was omitted as it only serves to direct the air
stream downward, and this effect could be otherwise simulated. The lower lip was included
because it acts as a vertical spacer between the air stream and the flute’s embouchure hole.
The completed flute-playing device is shown in Figure 21.
20
Figure 21: Completed Flute Playing Device on a PVC Flute
Air is blown into the system using a pressure source in the lab (Higgins Labs Room 031).
Pressure and volume are controlled using a regulator and two cutoff valves: one located on the
regulator and the other on the lab’s air nozzle. Air then enters a tube that connects to the
perturbation chamber, which is crafted from a prescription pill container. In the chamber, the
air stream is disturbed in a manner comparable to the human mouth. Another tube then
passes the air over a rolled piece of paper, which acts as the simulated lower lip. Finally, the air
passes over the embouchure hole and sounds the desired note.
The flute playing device was hooked up to the regulator and the pressure source at the
beginning of each testing period. The flute was then attached to flute playing device’s tension
clamp. The flute’s tone holes were taped until the proper fingering for the desired note was
attained. The position of the clamp on the flute was adjusted until the flute sounded the
desired note.
21
Testing Previously Developed Equation Models
At the very beginning of the mathematical modeling, the group tested out the equation
models that were developed by previous musical scholars and researchers. Various
mathematical equations were found for musical instruments focusing on the simple flutes. The
group tested two mathematical equations that were applicable to this particular project.
Chris Forster, a musical instrument builder, composer and scholar, developed a series of
mathematical equations for flute construction based on Nederveen’s book Acoustical Aspects
of Woodwind Instruments in 1969. The equations listed below were tested on the initial rapid
prototyped flute. The flute was played both by the blowing rig and human players, and
frequencies were recorded for calculations of the tone hole locations with all the tuning slides
fully shortened.
1
2)(*45.0
d
ddlL H
HHH
eeb
emb HWD
dC *)
*
4(
2
The embouchure correction value, which is the distance from embouchure hole to the
left of the cork, was also calculated by using the equation given in the book Fundamentals of
Musical Acoustic for three different types of flutes. Listed parameters of the flute were
measured as well as the lip thickness of the player for this calculation. The results of the testing
of these equations are reported in the previously developed equations section under Results.
Initial Mathematical Modeling
Equations that the group developed for determining the hole placements for the Irish
flute were based on the principle of a tube with two open ends. Correction lengths for both
ends were applied in order to determine the tone hole placement.
The first correction was at the embouchure. The embouchure correction length (Δle)
accounted for that end of the tube being stopped, as well as the existence of the embouchure
hole on the cylindrical surface of the flute. This correction was used in all of the tone hole
calculations.
22
At the other end of the flute, there were two different corrections depending on the
number of the tone holes that were covered. A correction length for the end of the flute
(Δlhemb) was used when all the tone holes were closed on the flute. This correction accounted
for the sound wave leaving the end of the tube. When any of the tone holes were open, the
correction length for the tone hole (Δlh) was used. This correction adjusted for the sound wave
leaving through a hole along the cylindrical face of the flute as well as the size of that hole.
The wavelength is equivalent to the length of the tube, open at both ends, needed to
produce the desired frequency. In the equations to follow, this length is referred to as Ls.
Correction factors are subtracted from the wavelength. With these corrections removed, the
distance from the center of the embouchure hole to center of the tone hole (La) is determined.
In the case where all the tone holes are closed, this length is equal to the distance from the
embouchure hole to the end of the flute. A simple visual of these correction lengths can be
seen in Figure 22.
Figure 22: Correction Length used in Determining Tone Hole Location
The equation for the correction lengths that the group created were based on five basic
observed relationships. The first relation states that if the diameter of the embouchure hole
(de) is equal to that of the diameter of the bore (dbe) at the embouchure, then the correction
length at the embouchure (Δle) is equal to zero. If the two diameters are the same size, this is
equivalent to cutting the bore of the flute off at this point.
1. If de=dbe, Δle=0
This same principle is seen in the second relationship. The relationship states that if the
diameter of the tone hole (dh) is equal to the diameter of the bore at the tone hole (dbh) then
23
the correction length at the tone hole (Δlh) is equal to zero. The reasoning for this is the same
as was the reasoning for the embouchure hole.
2. If dh=dbh, Δlh=0
The third and fourth relations state that if the wall thickness at the tone hole (h1)
increases or the wall thickness at the embouchure hole increases, then the frequency will
decrease. The longer the pipe used to produce a sound, the lower the pitch. Therefore,
increasing the wall thickness will result in a lower pitch.
3. If h1 increases, frequency decreases
4. If h0 increases, frequency decreases
The final relation states that if the diameter of a tone hole (dh) decreases, then the
frequency produced will also decrease. The inverse of this relation is also true.
5. If dh decreases, frequency decreases
As stated in the second relation, the correction length at the tone hole becomes zero
once the tone hole diameter equals that of the bore diameter at the tone hole. The smaller the
tone hole, the further the sound wave node will form past that tone hole. This creates a longer
wave which produces a lower frequency. This change in the node placement due to a decrease
in hole diameter can be seen in Figure 23.
Figure 23: Change in Wavelength Node due to Reduction in Diameter
24
This causes the theoretical length of the pipe to be longer than the actual distance from
embouchure hole to the tone hole, thus lowering the frequency.
The equations that the group developed were based off the five relations stated above.
Dimensional analysis was also used to ensure that the units worked out. The equations took
the wavelength of a desired frequency and subtracted corrective lengths from both ends of the
flute in order to compensate for the flute’s embouchure and tone holes. The following
describes the equations that the group developed.
When determining hole placements for a flute, it was difficult to know the diameter of
bore at the tone hole without first knowing the location of the tone hole. This situation
required a method of estimating the diameter of the bore. This estimate for the bore diameter
at the tone hole (dest) was calculated using the formula seen below.
end
b
dendbeest d
F
Fddd
2*1*
22
This equation uses the principle of similar triangles in order to estimate the diameter of
the bore. Using half of the diameter of the bore at the embouchure hole (dbe) and half of the
diameter of the bore at the end of the flute (dend) will result in a triangle that can be used to
determine dest. In order to find the rough length of where the hole should be, the ratio
between the desired frequency (Fd) and the base frequency (Fb) is used. Because the angle of
taper of the bore is not large this approximation produces fairly accurate results.
This same “similar triangle” technique was used to estimate the wall thickness (hest).
This results in the equation as well as the dimensional analysis is seen below.
mmHz
Hzmmmmmm
hF
Fhhh end
b
dend
est
2*1*22
2*1*22
0
The equation uses the wall thickness at the embouchure hole and the wall thickness at
the end of the flute to estimate the wall thickness at the tone hole in question.
The correction length at the embouchure hole (Δle) was calculated using the ratio of the
differences between the area of the bore of embouchure (Abe) and the area of the embouchure
25
hole (Ab) over the diameter of the bore at the embouchure hole (dbe) minus the diameter of the
embouchure hole (de). This value was then added to height of the wall thickness at the
embouchure (ho). This equation is shown below as well the dimension analysis for this
equation.
mmmmmm
mmmmmm
hdd
AAl
ebe
ebee
)(
)(
)(
)(
22
0
An equation was needed to determine the correction length at the end of the flute
when all of the tone holes were closed. This equation uses the ratio between the base
frequency (Fb) and the desired frequency (Fd) added to 5 times the wall thickness at the end of
the flute (hend) and that same ratio times the bore diameter at the end of the flute (dend). The
equation as well as the dimensional analysis can be seen below.
mmHz
Hzmm
Hz
Hzmm
dF
Fh
F
Fl end
d
bend
d
bhemb
**5
**5
The equation to determine the correction length for the tone holes was determined
using the ratio between the difference of the estimated area of the bore (Aest) and the area of
the active tone hole (Ae) over the estimated diameter of the bore at the active tone hole (dest)
minus the diameter of the tone hole (de). This ratio was multiplied by the ratio of the base
frequency (Fb) and the desired frequency (Fd) as well as the ratio of the estimated diameter of
the bore at the active tone hole (dest) over the diameter of the tone hole (dh). Added to this
was 5 times the estimated wall thickness (hest) and 0.5 times the ratio of the base frequency (Fb)
and the desired frequency (Fd) times the estimated diameter of the bore at the active tone hole
(dest). The estimate area of the bore was the calculated using the estimated bore diameter.
This equation for the calculation of the correction fact at an active tone hole as well as the
dimensional analysis of the equation can be seen below.
26
mmHz
Hzmm
mm
mm
Hz
Hz
mmmm
mmmmmm
dF
Fh
d
d
F
F
dd
AAl est
d
b
est
h
est
d
b
hest
hest
h
**5.0*5**)(
)(
**5.0*5**)(
)(
22
The distance from the embouchure to the active tone hole (La) was determined by
taking the actual length of the desired frequency wavelength (Ls) and subtracting the correction
length for the embouchure (Δle) as well as the corrective length for the active tone hole (Δlh).
The distance from the embouchure to the end of the flute (Laemb) was determined by
taking the actual length of the desired frequency wavelength (Ls) and subtracting the correction
length for the embouchure (Δle) as well as the corrective length for the end of the flute (Δlhemb).
All of the equations that the group developed and discussed above can be collectively
below.
hembesaemb
hesa
est
d
b
est
h
est
d
b
hest
hest
h
end
d
b
end
d
b
hemb
ebe
ebe
e
end
b
dend
est
end
b
dendbe
est
llLL
llLL
dF
Fh
d
d
F
F
dd
AAl
dF
Fh
F
Fl
hdd
AAl
hF
Fhhh
dF
Fddd
**5.0*5**)(
)(
**5
)(
)(
2*1*22
2*1*22
0
0
The equations above do not take into account the chimney height above the outer bore
(hc). This was considered after these equations were applied to determine the hole placement.
If the distances between the tone holes were too far for fingers to fit comfortably, then the
distance between them could be decreased and the chimney height of the affected holes above
the bore could be increased.
27
In order to easily compute values for the placement of tone holes, all the necessary
values and equations were entered into a spreadsheet which can be seen Figure 46 in Appendix
A. The values highlighted in orange are the input values needed for the calculations. The
values highlighted in green are the final calculated values for the tone hole placement. This
allowed all the values for each tone hole to be computed simultaneously. The spreadsheet
enabled data from different tests to be easily entered and analyzed.
Frequency Gathering of PVC, Delrin®, RP, Olive, and Mopane Material Flutes
In order to refine and confirm the equations that were developed, a number of different
tests were performed to obtain experimental data. Some of these were used in order to
determine whether or not certain changes to the flute would greatly alter the frequencies. Six
different flutes were used in these tests. The primary flutes used were a D flute that was made
out of ¾ inch PVC pipe, a D flute that was made out of Delrin®, and the group’s initial rapid
prototyped flute that was based on the Delrin® Flute.
Human Player Frequency Testing
Each note of the flute was played in the two octaves in question. The upper octave of
the flute was played with the same fingerings as the lower octave, but overblown. While one
group member played the flute, another group member measured the frequencies of the notes
played. Frequency was measured using the smart phone tuner application “gStrings”. A screen
shot of this application can be seen below in Figure 24.
Figure 24: gString Tuner Application
28
The frequencies were then recorded either manually or in Excel. Human frequency
testing was performed on the Delrin®, PVC, RP, Mopane, and Olive flutes. All flute testing was
done with the tuning slides pushed fully in. This was to ensure consistency throughout the
group’s testing.
Flute Playing Device
The flute playing device was hooked up to the regulator and the pressure source at the
beginning of each testing period. The flute was then attached to flute playing device’s tension
clamp. The flute’s tone holes were taped until the proper fingering for the desired note was
attained. The position of the clamp on the flute was adjusted until the flute sounded the
desired note. Frequency was then measured and recorded. Frequency testing with the flute
playing device was performed on the rapid prototyped flute only.
Chimney Height
To vary the chimney height, the group used a series of 15 different cardboard sheets,
each with holes cut to fit the size of the flute’s tone holes. This set up can be seen below in
Figure 25.
29
Figure 25: Set up using Cardboard to Increase Chimney Heights
During testing a clamp was used to apply pressure to the cardboard to hold it against
the flute. Another technique that was used was applying layers of tape to increase the chimney
height and then drilling through to open up the tone hole. Frequencies were measured and
recorded in the same manner as for human frequency measurements. Instead of one set of
data, however, five sets of data were taken: each with a different number of cardboard sheets
wrapped around the bore at each tone hole. The height of the cardboard sheets above the
tone hole was measured for each trial using a caliper. Each note-chimney height combination
was played three times and the average of the frequencies was taken. Chimney height testing
was performed on the rapid prototyped flute only.
Bore Extension Testing
Bore extension testing was performed in much the same manner as was the frequency
testing. For this test the bore was extended by certain increments and frequency data was
taken for two octaves. There were a total of four different extension amounts including a base
30
set of data with all the tuning slides pushed in. Bore extension testing was performed on the
rapid prototyped flute only.
Temperature
Temperature data was taken before the frequency and chimney height tests were
performed. A digital thermometer was used to record ambient temperature, as well as the
temperature at the far end of the bore, inside each tone hole, and inside the embouchure hole.
The flute was warmed up by playing before these tests were performed, and all temperature
values were measured while the flute was being played.
Cork Displacement
Cork displacement testing was performed in much the same manner as was the human
frequency testing, except that five sets of data were gathered, and each of these data sets
corresponded to a different tuning cork position. Before each data set was taken, the distance
from the cork to the end of the bore at the first joint was measured by inserting a wooden
dowel into the flute and drawing a line on the dowel at the first joint. The distance between
the end of the dowel and the line was then measured using a tape measure. Cork displacement
testing was performed on the Delrin® flute only.
Human Player Averaging
In human player averaging, frequency data was taken over two octaves for four
different human players, one of whom was not member of the group. The frequency data
points of these players were then averaged to ensure that the group’s flute players played in-
tune.
Final Rapid Prototyping
For the final rapid prototype the major change was to increase the E tone hole and
increase the chimney height so that the hole could be moved closer to its neighboring tone
holes. The decision was also made not to print a cork into the flute, but to insert a cork after
the flute was printed. This gave the flute more versatility when it came to tuning the second
octave. The tone holes, the locations of which were determined by the set of equations, were
31
moved in closer to the embouchure hole by 10 mm. This was done so that the flute would be in
tune with the tuning slide pulled out 10mm. This allows the player to push the slide in if
atmospheric conditions are causing the flute to play flat. A cross section of the flute along with
the locations of the tone holes can be seen in Figure 26.
Figure 26: Final Rapid Prototype Cross Section with Tone Hole Locations
The chimney height on the last tone hole was increased by 4.87mm. This was done to
compensate for increasing the diameter of that hole to 8mm. If the chimney extension was not
added that hole would have been placed an additional 15mm away from the previous hole
making it difficult to play.
The flute was printed horizontally to reduce the number of pieces in the flute to three.
All six tone holes where placed on one piece to eliminate any shear loads on a joint that was in
between sets of tone holes. A joint was included after the tone holes where a small section of
the flute attaches to complete the flute’s length. This overlapping joint contains two small rings
around the smaller joint section. These allow the overlying section to have only two points of
contact. This was done in an effort to eliminate any rocking in the joint. A close-up of this joint
can be seen below Figure 27.
32
Figure 27: Close of End Joint in Final Flute
The joint closest to the embouchure hole was extended to overlap by 50 mm to
accommodate for tuning. This joint was left smooth without any contact rings so that the joint
can be easily extended. This removed the possibility that the upper part of the joint will fail to
make contact with a contact ring and become unstable.
The final rapid prototype was also created using an Objet260 Connex Rapid Prototype
machine. The material used was VeroWhite. The .stl files were saved with a tolerance of
0.1524 mm. The group chose to use the finest tolerance in order to reduce any effects of
surface roughness on the model. A gap of 0.0254 mm was used between the joints.
Contact Rings Contact Rings
33
Results
Results of Initial Prototype
The group’s initial rapid-prototyped flute was playable with relative ease. In terms of
the finish, this flute was noticeably rougher than the any of the wood or Delrin® flutes that
were encountered. For the most part the 0.0254 mm gap that was designed into the joints of
the flutes resulted in a tight fit. There were a couple of joints that had a loose fit. In order to
seal these joints the group used Teflon tape. This not only helped to seal the joints but added
more rigidity to the flute as a whole. The decision to print a cork in a fixed position was one
that limited some of the desired tests. One test was the cork displacement so not being able to
move the cork position to see the influences was a limitation for the analysis in the rapid
prototype.
Upon close examination of rapid prototyped flute it was seen that the layering effects
from the printer are very small and are not very noticeable or intrusive. These layers can be
clearly seen in Figure 28.
Figure 28: Magnification of Layers Created by Rapid Prototyping Vertically
34
The decision to make the flute in six pieces turned out to have an effect that the group
was not expecting: the multiple joints resulted in a flute that was not a rigid as the flutes that
the group used for a comparison.
A final concern of the rapid prototyping process was the cost of manufacturing. The
group’s first flute cost $378.43 in materials. If manufacturing processes were considered the
total price would increase.
Previously Developed Equations
Based on Forster’s formula, effective length of the tone hole could be calculated by the
formula shown below. In this equation lH is the bore thickness at the tone hole, dH is the tone
hole diameter, and d1 is the bore diameter at the tone hole.
1
2)(*45.0
d
ddlL H
HHH
Table 1: Testing of Forster's Effective Length of Tone Hole Equations
Note Calculated Distance from Tone Hole to Embouchure (mm)
Measured Distance from Tone Hole to Embouchure (mm)
Difference (mm)
% Error
D4 553.2 517.9 35.3 -6.38
E4 459.22 424.89 34.32 -7.48
F#4 414.65 386.64 28.01 -6.76
G4 382.79 358.02 24.77 -6.47
A4 315.87 298.02 17.85 -5.65
B4 285.83 263.93 21.9 -7.66
The formula was tested on the RP flute and a human player played both octaves. As
shown in the result chart, the difference between the calculated distance from tone hole to
embouchure hole and the actual measured length was considerably large, ranging from 17.8
mm to 35.3 mm in the first octave and even greater in the second octave. Several attempts
were done to re-derive the original equation and adjust it by modifying the coefficients.
However, all the attempts only resulted in minor changes. Therefore, the team concluded that
the Forster equation was not able to correctly predict the RP flute tone hole distance from the
embouchure hole based on the given parameters.
35
As previously indicated, based on the book Fundamentals of Musical Acoustics, the
distance from embouchure hole to the left of the cork can be determined from the following
formula:
eeb
emb HWD
dC *)
*
4(
2
Table 2: Calculation of Distance from Embouchure Hole to the Left of the Cork
Flute Cemb(mm) RP 32.70
PVC 217.29
African black wood 37.05
The book clearly indicates that most flutes have a Cemb value around 50 mm. The
equation was put to the test and yielded results of 33 mm and 37 mm for the RP flute and the
African black wood flute accordingly. The PVC flute yielded a very large value for Cemb based on
this equation due to its relatively large radius of air column and small chimney height. Even
though there could be measurement uncertainty, especially on the lip thickness of the human
player, the differences were quite large considering the magnitude of the total distance of the
flute.
Uncertainty Calculations
The final uncertainty of the group’s equations was determined for each hole location on
the flute. The uncertainties for each of the parameters were based on the accuracy of the
group’s measurements. For the group’s diameter measurements the group’s calipers operated
with an uncertainty of 0.01 mm. The group’s length measurements had an uncertainty of 0.1
mm because a tape measure was required to measure longer distances. This resulted in a
range in relative uncertainty from 0.02% to 0.24% in calculating each hole location relative to
the embouchure hole. This can be seen in Table 3. The higher-percent uncertainties were
found in the upper octave. This is because the higher frequencies have a greater influence on
the uncertainty in the equation.
36
Table 3: Relative Uncertainty in Tone Hole Placement Calculations
Note Frequency
Desired
Tone Hole
Placement
Uncertainty
D4 293.66 0.02%
E4 329.63 0.04%
F#4 369.99 0.04%
G4 392 0.04%
A4 440 0.06%
B4 493.88 0.07%
C#5 554.37 0.10%
D5 587.33 0.02%
E5 659.26 0.07%
F#5 739.99 0.08%
G5 783.99 0.10%
A5 880 0.13%
B5 987.77 0.18%
C#6 1108.73 0.24%
D6 1174.66 0.02%
General Frequency
The results of the general frequency test were in keeping with the group’s expectations.
This test was performed with all the tuning slides pushed in. As seen in Figure 29, Figure 30,
and Figure 31, in general, the flutes were sharp which can be seen in the positive difference
value. The figures below show the difference between the played frequency and the actual
frequency in relation to the actual frequency. All of the data for these tests can be seen in
Figure 47, Figure 48, and Figure 49 in Appendix A.
37
Figure 29: Difference in Frequency vs. Actual Frequency for the Mopane Wood D Flute
Figure 30: Difference in Frequency vs. Actual Frequency for the Olive Wood D Flute
y = 0.0297x - 6.5233 R² = 0.8438
0.000
5.000
10.000
15.000
20.000
25.000
30.000
35.000
0 200 400 600 800 1000 1200 1400
Ave
rage
Fre
qu
ency
min
us
Act
ual
Fre
qu
ency
(H
z)
Actual Frequency (Hz)
Difference in Frequency vs. Actual Frequency- Mopane Wood D Flute
y = 0.0306x - 8.7773 R² = 0.8014
-5.000
0.000
5.000
10.000
15.000
20.000
25.000
30.000
0 200 400 600 800 1000 1200 1400
Ave
rage
Fre
qu
ency
min
us
Act
ual
Fre
qu
ency
(H
z)
Actual Frequency (Hz)
Difference in Frequency vs. Actual Frequency- Olive Wood D Flute
38
Figure 31: Difference in Frequency vs. Actual Frequency for the Olive Wood Eb Flute
Cork Displacement
The results of the cork displacement test can be seen in Table 4. The effect of the cork
displacement appears to be negligible in the first octave. However in the upper octave the
influence was much more prevalent. This is very noticeable in Table 5 which shows the change
in frequency based on the position of the cork. Values that appear negative in Table 5 mean
that the frequency actually increased when the cork was moved further away from the
embouchure hole. However, these variations from the trend could account for in the fact that
the change is very small. This deviation could be attributed to variations in the player’s
embouchure during testing. Before testing the cork was at 16.78 mm away from the edge of
the embouchure hole. A graph of the change in frequency from the base frequency for
different cork positions can also be seen in Figure 32.
y = 0.0098x - 8.078 R² = 0.2182
-15.000
-10.000
-5.000
0.000
5.000
10.000
15.000
20.000
0 200 400 600 800 1000 1200 1400
Ave
rage
Fre
qu
ency
min
us
Act
ual
Fre
qu
ency
(H
z)
Actual Frequency (Hz)
Difference in Frequency vs. Actual Frequency- Olive Wood Eb Flute
39
Table 4: Frequency Played Relative to Cork Position
Table 5: Change in Frequency with Respect to Cork Position
0 10.3 16.78 31.42 36.61
D4 296.7 297.7 294.2 294.7 292.5
E4 335.8 337.9 334.6 336 333.3
F#4 377.5 377.7 373.3 377.2 371.6
G4 403.2 404.2 396.7 402.6 395.5
A4 456.6 456.7 451.2 454.4 448.8
B4 510.7 514.9 501.3 510.7 499.7
C#5 565.3 566.6 554.3 566.3 547.1
D5 599.2 600.3 593.2 596.4 588.8
E5 680 682.4 672.1 670.4 660.2
F#5 764.6 762.2 751.2 748.2 737
G5 815 811.4 799.9 800.1 784.2
A5 919.2 912.3 896.6 897.3 872.1
B5 1042 1033 1006 995.6 964
C#6 1151 1146 1111 1080 1040
D6 1234 1224 1202 1182 1144
Fre
qu
en
cie
s p
laye
d(H
z)
Distance from Edge of Embouchure Hole to Cork (mm)
10.3 16.78 31.42 36.61
D4 -1 2.5 2 4.2
E4 -2.1 1.2 -0.2 2.5
F#4 -0.2 4.2 0.3 5.9
G4 -1 6.5 0.6 7.7
A4 -0.1 5.4 2.2 7.8
B4 -4.2 9.4 0 11
C#5 -1.3 11 -1 18.2
D5 -1.1 6 2.8 10.4
E5 -2.4 7.9 9.6 19.8
F#5 2.4 13.4 16.4 27.6
G5 3.6 15.1 14.9 30.8
A5 6.9 22.6 21.9 47.1
B5 9 36 46.4 78
C#6 5 40 71 111
D6 10 32 52 90
Ch
ange
in F
req
ue
nci
es
fro
m B
ase
Te
st (
Hz)
Distance from Edge of Embouchure Hole to Cork (mm)
40
Figure 32: Change in Frequency Due to Cork Position
Bore Extension
The summarized results of the bore extension tests can be seen in Table 6 below. This
testing was performed on the rapid prototype flute only. The change in frequency between the
extended frequency and the base frequency of no extension was calculated and can be seen in
Table 7.
41
Table 6: Results from Bore Extension Test
Table 7: Difference in Frequency from Base
This change in frequency was then divided by the base frequency of each tone hole (Fbt)
and plotted against the same base frequency. A trend line was then found for this set of points.
0 3.82 7.03 10.52
D4 289.1 288.4 285.4 284.5
E4 328.6 326.6 322.8 320.9
F#4 369.9 364.7 362.8 361.9
G4 396.5 391.2 386.9 385.4
A4 447.3 442 440 434.4
B4 505.3 500.1 493.6 486.7
C#5 570 555.5 550.1 544.4
D5 598.3 591.3 588 586.1
E5 673.3 667.7 659.2 658.6
F#5 759.4 751.3 742.5 741.2
G5 812.2 802.7 795.5 797
A5 911.1 900.1 888.8 888.7
B5 1020 1010 998.7 997.2
C#6 1122 1110 1095 1091
D6 1203 1188 1182 1178
Distances (mm)
Freq
uen
cies
(H
z)
3.82 7.03 10.52
D4 0.7 3.7 4.6
E4 2 5.8 7.7
F#4 5.2 7.1 8
G4 5.3 9.6 11.1
A4 5.3 7.3 12.9
B4 5.2 11.7 18.6
C#5 14.5 19.9 25.6
D5 7 10.3 12.2
E5 5.6 14.1 14.7
F#5 8.1 16.9 18.2
G5 9.5 16.7 15.2
A5 11 22.3 22.4
B5 10 21.3 22.8
C#6 12 27 31
D6 15 21 25
Freq
uen
cy D
iffe
ren
ce f
rom
Bas
e (H
z)
Distances (mm)
42
This trend line was found to be approximately a linear function with the exception of B and C#.
The graphs along with their associated trend lines can be seen in Figure 33, Figure 34, and
Figure 35.
Figure 33: Change in Frequency over the Base Frequency (Fbt) with Respect to the Base Frequency (Fbt) for a Bore Extension of
3.82 mm
Figure 34: Change in Frequency over the Base Frequency (Fbt) with Respect to the Base Frequency (Fbt) for a Bore Extension of
7.03 mm
y = 2E-06x + 0.0101 R² = 0.014 0
0.005
0.01
0.015
0.02
0.025
0.03
0 200 400 600 800 1000 1200 1400
Ch
ange
in F
req
uen
cy o
ver
Bas
e
Freq
uen
cy o
f To
ne
Ho
le
Base Frequency of Tone Hole (Hz)
Extension of 3.82 mm
y = 3E-06x + 0.019 R² = 0.0336 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 200 400 600 800 1000 1200 1400
Ch
ange
in F
req
uen
cy o
ver
Bas
e F
req
uen
cy
Base Frequency (Hz)
Extension of 7.03 mm
C#
C#
43
Figure 35: Change in Frequency over the Base Frequency (Fbt) with Respect to the Base Frequency (Fbt) for a Bore Extension of
10.52 mm
The slope of this line was very close to zero and therefore was approximated as such.
The equation was solved for the change in frequency. This equation was then repeated for the
other two bore extensions. These equations are the following:
bt
bt
bt
ff
ff
ff
0272.0
019.0
0101.0
The coefficients in these equations were plotted against the extension distance (Ex).
This linear equation was then plugged back into the starting equation which resulted in:
btx fEf *)00009.0*0024.0(
This equation holds fairly true to all the scale degrees except for B and C#. To correct
for this, the previously-stated equation was turned into a step function. This function corrects
for the fact that these two tone holes resulted in a greater change in the frequencies. This step
function can be seen below.
y = -3E-06x + 0.0272 R² = 0.0121 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0 200 400 600 800 1000 1200 1400
Ch
ange
in F
req
uen
cy o
ver
Bas
e F
req
uen
cy
Base Frequency (Hz)
Extension of 10.52 mm
C#
B
44
5804802
5804801**)00009.0*0024.0(
bt
bt
btxf
ffEf
i f ,
i f ,
This equation states that if the base frequency of the tone holes is in between 480 Hz
and 580 Hz, then multiply the equation by a factor of 2. If not, then the equation will be
multiplied by a factor of 1. This compensates for the B and C# having a greater change in
frequency.
There are small differences between what was predicted and the actual frequencies
played. These can be seen in Figure 50 in Appendix A. Some of this error may be explainable
due to inconsistencies in playing and with further testing these errors could be reduced. The
equation stated above is independent of the equations used in determining the design of the
final model.
Temperature
Figure 36 shows the results for temperature testing. Temperature testing was
performed on the rapid prototyped flute only. The temperature probe was inserted into each
of the specified holes as the flute was being played. As expected, the temperature increases as
the reading point approaches the flute’s embouchure hole. The temperature did drop off quite
rapidly after the first few tone holes. Using this data, an average temperature of 25 °C was
used to determine the speed of sound, and this temperature is used for tone hole placement.
Figure 36: Temperature Variations within the Bore of the Flute while being Played
Ambient
D4 E4 F#4 G4 A4 B4
C#5
Embouchure
20
22
24
26
28
30
32
34
Tem
per
atu
re (
°C)
Hole
Temperature Variation within Flute
45
Human Player Averaging
Human player averaging tests were performed on the rapid prototyped and Delrin®
flutes. The summarized results are shown below in Figure 37 and Figure 38. Ideally, the
difference between the played frequency and the actual frequency should be zero. However,
the all of the tuning slides where pushed in, which resulted in the flute being sharp. This causes
the negative difference between the two frequencies. For the rapid prototype, the average
played frequencies continued to get sharper as frequencies increased. The Delrin® Flute, on
the other hand, stayed relatively sharp the through the entire range of the scale. In the rapid
prototyped flute the higher frequencies could be brought more in tune by moving the cork
further away from the embouchure hole. The entire flute can then be tuned by adjusting the
flute’s main tuning slide.
Figure 37: Difference between Played Frequency and the Actual Frequency in Average Human Player Test on the Initial Rapid
Prototyped Flute
y = -0.0346x + 8.7883 R² = 0.7146
-35
-30
-25
-20
-15
-10
-5
0
5
0 200 400 600 800 1000 1200 1400
Act
ual
Fre
q-A
vg F
req
(H
z)
Actual Frequency (Hz)
RP: Actual Freq-Avg Freq vs. Actual Freq
46
Figure 38: Difference between Played Frequency and the Actual Frequency in Average Human Player Test on the Delrin®
Flute
Chimney Height Results
Predicting Frequency
Based on the data collected during experiments by using cardboard to increase the
chimney height, a fairly accurate frequency prediction function was found. Matlab was used to
draw a 3-D graph for determining the relationship among three values: tone hole distance to
embouchure, increased chimney height, and frequency played. The 3-D graph (Figure 39)
shows a surface with a nearly linear relationship among the values. Therefore, by using a least-
squares and linear regression approach, a linear function was found:
0704.809*3175.1**1546.1 ghtChimneyHeiDistanceFrequency
The average difference between predicted frequency and calculated frequency was 8.57
Hz and the average percent error was 1.98%.
y = -0.0141x + 2.9483 R² = 0.2481
-35
-30
-25
-20
-15
-10
-5
0
5
0 200 400 600 800 1000 1200 1400
Act
ual
Fre
q-A
vg F
req
(H
z)
Actual Frequency (Hz)
Delrin: Actual Freq-Avg Freq vs. Actual Freq
47
Figure 39: 3-D graph showing relationship among distance, chimney height and frequency
The frequency data tested at zero chimney height (no cardboard added) could be
eliminated since this value was already calculated using the frequency equation that the group
developed. In doing so, the final function achieved a higher accuracy prediction rate, which
lowered the average difference between predicted frequency and calculated frequency to 8.40
Hz and the average percent error to 1.95%. The final frequency-predicting equation is shown
below:
463.805*8875.0**152.1 ghtChimneyHeiDistanceFrequency
Predicting Chimney Height
A similar linear regression approach was used to predict chimney height based on a
given frequency and the distance between each tone hole and the embouchure hole. However,
the results had a large error and did not yield predicted chimney height within expectations.
The relationship among the three parameters did not demonstrate linear behavior or a
predictable pattern.
Several factors could have contributed to this unsuccessful prediction. First of all, due
to the time constraint, not enough data was taken. This made it difficult to observe a trend in
the data. Secondly, there may have been inaccuracies in the collected data. This could be
48
improved by using both a human player and the flute playing device, and averaging the
frequencies from the two otherwise-identical tests.
As there was not enough time to determine an equation for predicting the effects on all
the chimney heights, it was decided to use the results from our E tone hole to determine a
suitable chimney height for our final model because it demonstrated a consistent linear trend.
The graph of the results for the E tone hole as well as a trend line used to calculate the chimney
height on the final rapid prototype can be seen in Figure 40.
Figure 40: Frequency Change due to Chimney Height Increase in the E4 Tone Hole
Flow Inside the Bore
An important consideration was the degree of turbulence of the flow in the bore. To
ascertain a turbulence reading, the Reynolds number of the flow through the bore was
calculated according to the following equation:
Although the group did not have the equipment to measure the velocity inside the bore,
the group was able to use an anemometer to measure airstream velocity at the embouchure
hole. Since the velocity inside the bore was known to be lower than the velocity at the
embouchure hole, the measurement at the embouchure hole presented a valuable “worst case
y = -0.512x - 0.7038 R² = 0.9079
-2
0
2
4
6
8
10
-18 -16 -14 -12 -10 -8 -6 -4 -2 0
Ch
imn
ey H
eigh
t In
crea
se (
mm
)
Frequency Difference (Hz)
Chimney Height Increase vs. Frequency Difference for E4
49
scenario” velocity. The velocity was found to be 4.47 m/s. This, paired with other known
values, produced a Reynolds number value of:
This value placed the flow just inside the turbulent range of values. Laminar flows occur when
Re < 2300 and turbulent flow occurs when Re > 4000. The range in between this is defined as
transitional flow, where both laminar and turbulent flows are possible. However, since a large
value for velocity was chosen, the flow could realistically be placed in the transition range.
Implementation of the Equations for Manufacturing
The group’s equations were implemented in an Excel document to facilitate the
computing of the tone hole locations. The equations that the group used were difficult to use
in the equation editor within SolidWorks as they required a set of approximately 16 different
variables for each tone hole. This made it necessary to find another way to create a link
between the calculation sheet and the SolidWorks model. This was achieved by creating a
design table in SolidWorks, which is also Excel based, and linking the values between the two
different spreadsheets. To aid in the linking of the values, the dimensions within SolidWorks
were renamed for easy identification within the design table. When this method is used to link
the spreadsheets, only the final dimensions needed for the model are in the design table.
Whenever the calculation spreadsheet was updated, all that was necessary to update the
model was to open the design table within SolidWorks and then close it.
It was later discovered that the entire calculation spread sheet could be copied and
inserted into the design table after making sure to skip a few lines in the design table. This was
to ensure that SolidWorks did not try and assign the calculations as new configurations. A small
sample of this can be seen in Figure 41.
50
Figure 41: Calculations Inserted into Design Table
Developed equations
and variables need to
determine tone hole
placement
Dimensions found in flute model
Values of dimensions
that are linked to the
below equations
51
The final results of the calculations were referenced in the default configuration within
the design table. The input values could then be changed directly in the design table and the
updated values would propagate directly into the model.
In terms of manufacturing, a link between the equations governing a model and the
model itself was beneficial. This strategy facilitated modifications to the input dimensions, such
as hole diameters, and the model could be easily updated to reflect this change. This process
streamlined custom fabrications without the user having to manually change values or create a
whole new model. In a work environment, since this model already contains the equations, it
would be very easy to share the model without having to worry about losing the calculation
worksheet or breaking any links.
Results of Final Prototype
The final prototype fulfilled the goals of the project in that the measured frequencies
obeyed the A440 standard closely. After extensive cleaning, the flute proved to be easily
playable, and produced a clear tone. The hole spacing left some room for improvement, as
fingering the last two tone holes proved more difficult than expected. This difficulty would
have been corrected if the holes had been angled on the bore to better suit average human
finger length ratios. The final model of the flute can be seen below.
Figure 42: Final Rapid Prototype
Printing the flute horizontally did not negatively affect playability. The layers created by
rapid prototyping can be seen in Figure 43.
52
Figure 43: Magnification of Layers Created by Rapid Prototyping Horizontally
The cost of the material for this flute was $535.35. This would increase if the cost to
produce the flute was accounted for. The increased expense was a consequence of more
support material needed in the bore to print the flute horizontally. In the initial prototype
there was no need for support material in the bore as the pieces were printed vertically. In the
final rapid prototype the support material was very difficult to remove from the bore of the
flute due to the lengths of the pieces as well as the water nozzle used to dissolve the support
material did not fit into the bore.
Frequency data were taken using this flute. Initially the flute tuning slide was pulled out
to 10 mm, per the design. Data from this test showed that the flute was relativity uniformly flat
in the lower octave. The tuning slide was pushed in slightly and 3 sets of data were taken. Also
data was taken with the tuning slide pushed all the way in so there would be a comparison
against the data taken for the other flutes. The average of these sets was taken and compared
to the theoretical frequencies. This data can be seen below in Figure 44.
53
Figure 44: Frequency Data for the Final Rapid Prototype
The first octave proved to be in-tune with the theoretical frequencies. The second
octave contained some sharp notes. The frequencies of the final rapid prototype differed from
A440 by a maximum of 4.1%. When another group member played the flute, the sharp notes
were mostly corrected.
Note Actual Freq Test 1 Test 2 Test3 Test 4 Test 5 Test 6 Test 7
D4 293.66 287.6 289 291.3 289.4 292.6 293.3 294.3 289.9 293.4 3.8 0.3 1.28 0.09
E4 329.63 324.5 328.2 328.3 326.7 332.1 333.3 333.6 327.7 333.0 1.9 -3.4 0.58 -1.02
F#4 369.99 365.6 370.2 370.2 367.7 374.2 376.3 377.8 369.4 376.1 0.6 -6.1 0.17 -1.65
G4 392 389.3 392.3 395.4 391.1 401.8 402.9 404.3 392.9 403.0 -0.9 -11.0 -0.24 -2.81
A4 440 433.1 435.6 440.2 435.6 447.6 446.3 449.6 437.1 447.8 2.9 -7.8 0.65 -1.78
B4 493.88 490 493.4 500 492.2 510.6 510.1 515 495.2 511.9 -1.3 -18.0 -0.27 -3.65
C#5 554.37 551.4 561.2 561.2 554 575.5 573.8 579.4 558.8 576.2 -4.4 -21.9 -0.80 -3.94
D5 587.33 597.7 600.2 604 595.4 605.4 604.8 611.3 599.9 607.2 -12.5 -19.8 -2.13 -3.38
E5 659.26 674.2 677.3 682.3 674.4 682.2 683.2 686.9 678.0 684.1 -18.7 -24.8 -2.84 -3.77
F#5 739.99 764.6 765.3 766.6 765.5 768.4 774.4 777.5 765.8 773.4 -25.8 -33.4 -3.49 -4.52
G5 783.99 814 811.5 820.3 816.7 817.8 822.3 827.6 816.2 822.6 -32.2 -38.6 -4.10 -4.92
A5 880 886.5 888.6 897.2 889 898.5 903.2 905.6 891.6 902.4 -11.6 -22.4 -1.32 -2.55
B5 987.77 986.4 985.6 989.2 982.4 1007 1010 1011 985.7 1009.3 2.0 -21.6 0.21 -2.18
C#6 1108.73 1082 1087 1087 1079 1101 1124 1117 1084.3 1114.0 24.4 -5.3 2.20 -0.48
D6 1174.66 1174 1177 1174 1176 1193 2108 1199 1175.7 1500.0 -1.0 -325.3 -0.09 -27.70
gap= 10 mm, cork= 17 mm from center of embouchure hole Average Deviation 1.36 4.30
gap= 8.81 mm, cork= 20.2 mm from center of embouchure hole
gap= 0 mm, cork= 20.2 mm from center of embouchure hole
RP2 D Flute Frequency Test
Frequency (Hz)
Difference between
Average Played Frequency
and Actual Frequency (Hz) % deviationAverage Frequency
54
Conclusions
Final Prototype Assessment
Testing of the final prototype revealed that the flute was largely in tune. Therefore, it
could be concluded that the derived mathematical relationships were accurate in predicting
dimensional parameters based on frequency. The flute was optimal in that it obeyed the A440
standard as closely as possible despite player subjectivity and manufacturing error.
Manufacturing Feasibility
In the assessment of the feasibility to rapid prototype flutes the group ran across some
concerns with this method. For one thing, the price to print these flutes was upwards of $400
per flute. There were also time costs for this procedure. The time to grow these flutes using
the machine that the group used was more than eight hours. This does not include the time it
took to clean out the support material from the inner bore as well as the finishing of the
surfaces. These add up to a lot of time spent on after-printing procedures.
Until there is a quicker and less expensive way of rapid prototyping that results in a
more finished product, it would appear that rapid prototyping might not be a cost effective
means of producing flutes. Rapid prototyping does insure constancy among flutes created as
well as leaving the option open to create customized flutes. Such customized flutes could have
different hole spacing based upon hand sizes or even bore diameters depending on the comfort
levels required by the user.
55
Recommendations and Future Work
Chimney Height Prediction
The chimney heights prediction based on distance from each tone hole to the
embouchure and the desired frequency was not very successful due to the group’s limited time.
There are several things on which future research could focus for developing a better functional
equation.
Firstly, more sample data should be collected from varies flutes. The lack of data was
one of the main disadvantages. Without enough data points, patterns were hard to follow and
trends became hard to find. Curve fits were very difficult with such few data points. Use of the
neural network method was attempted for developing the chimney height model. Seventy
percent of the data were used to do the “training” while 30% of the data were used for testing.
Despite the better prediction that the neural network produced compared to the linear
regression method, the results were still not within the desired precision. If enough data were
collected, the neural network may have been able to accurately predict the chimney height
based on given parameters. Secondly, in order to ensure the accuracy of data collection and to
eliminate the human errors as much as possible, it is recommended that data be taken using
both the flute playing device and human players as much as possible. Both parties should play
the same flute and the same note should be played more than 3 times, and averages should be
taken. There seemed to be some outliers in the data that the team collected. However, by
using Chauvenet’s Criterion, the team was not able to remove the outlier. Therefore, it would
be the best to use the above average in an attempt to avoid the outliers in the first place.
Embouchure Hole
Future research may focus on the different possible shapes and sizes of embouchure
holes. The group explored elliptical embouchure holes as well as rounded rectangular
embouchure holes. While the African Blackwood, Mopane, and Olive wood flutes dealt with
elliptical shapes, the Delrin® and Rapid Prototyped had modern rectangular cuts which were
maintained in the final design because of the relative ease of playing the instrument. However,
56
the Mopane and Olive were also favorable to play. Future research on different cuts and
variations of the already-investigated holes could be explored.
Further research may also shed more light on the chimney height of the embouchure
hole, as well as the undercutting of the embouchure hole. According to the chimney height
data of the tone holes, increasing the chimney height of the embouchure hole could decrease
the frequency of the notes played in the flute. Altering the embouchure hole chimney height
could make the flute more sharp or flat depending on the configurations of other parts in the
flute such as length, bore size, and hole spacing. Experimenting with the undercut edge could
also be beneficial to the strength and sound of the notes produced. Contouring or rounding the
edges could produce different effects on tone.
Within the studies using the flute playing device, lip size and placement had a great
effect on whether or not a note could even be produced. A person with thicker lips could
essentially be increasing the chimney height and produce a different sound than a person with
thinner lips. An embouchure hole specific to the player him- or herself based on lip size and
thickness could be applied to a flute. Future studies could also look at the area of the
embouchure hole with respect to a person’s lips and determine if a certain cut is more
preferable based solely on lip size.
Bore Profile Modification
The bore profile is another area that could further be researched. The group worked
with cylindrical and conical bore types, but small variations could have substantial effects on
tone, pitch, and volume. Changing the angle of the taper is one area that can be explored. The
taper could be increased or decreased, and combinations of cylindrical and conical bores could
exist in the same profile. In Figure 45 below, models a few configurations of bore profiles.
57
Figure 45: Bore Profiles with Varying Tapers (Hopkin)
Each profile has a tapered distance from the cork, while the taper of the rest of the bore
varies. Some even have an increasing taper where the inside diameter of the bore at the end of
the flute is greater than the inside diameter of where the taper changes near the embouchure
hole. Such a design would mimic that of a plosive aerophone. This is used to improve octave
tuning.
Using different lubricants and polishing material on the flute could help improve the
intonation. The inside of the bore could be polished in this respect. An exploration of how a
“scoop” (increase in bore diameter) before each tone hole may change frequency and/or
volume would be beneficial.
Studies of pressure waves in the embouchure and the bore may also be valuable. There
may be a generalization concerning the different amplitudes produced and those could be
compared to the modes of vibration. With such information, a bore could be created that
mimicked the wave itself. Certain areas could alter and optimize the wave to produce the
desired sounds or frequencies at each note. Since people apply different amounts of pressure
through the flute when playing, a bore profile specific to each person based on his or her
“pressure profile” could be produced.
The surface roughness of the bore could also be investigated further. By creating two
identical flutes made from PVC the effects of surface roughness could be tested. Using a wire
brush, the bore of one of the flutes could be roughened. Leaving the other flute untouched the
58
two flutes could be played and compared. If there was a significant difference in the intonation
then more research could be done to determine the best way of coating the flute to have a very
smooth finish.
Future Work on Bore Extension
The equation that was developed for the change in frequency of each tone hole based
on the extension of the bore is a large area for future work. More data would need to be taken
to refine the equation and determine if this generalization is accurate. The explanation of why
the B and C# tone holes have a different impact on the change in frequency as well as all the
other tests done need to be explored as well. The benefits of refining this equation would be
the ability to predict how much the bore needs to be extended in order to change the
frequencies. The use of this equation to predict the location of the tone holes might be
possible with future developments.
Future Work Based on Final Prototype
Possible improvements to the final flute design are mostly centered on player comfort.
The lower tone holes of the flute were somewhat difficult to finger. Future work could use
ergonomic principles to increase the comfort level of the right hand, possible through rotation
of the tone holes about the flute’s center axis and the rounding of the top surface of the last
tone hole.
Furthermore, the end joint of the flute required modification by the group to tighten the
fit. Future researchers may wish to investigate different fitting techniques for the joints, such
as machined metal inserts.
59
Work Cited
Ayers, R. D. "The Conical Bore in Musical Acoustics."Conical vs Cylindrical Bores. American
Journal of Physics, Vol. 53, No. 6, pgs 528-537, 1985.
<http://www.phy.mtu.edu/~suits/conecyl.html>.
Benade, A. H. (1990). Fundamentals of Musical Acoustics, 2nd ed., New York: Dover
Publications Inc.
Burns, Casey (2006). "Irish Wooden Flute FAQ."Irish Wooden Flute FAQ. N.p., 28 Apr. 2006.
Web. 30 Jan. 2013. <http://www.caseyburnsflutes.com/faq.php>.
Cocchi, Alessandro & Tronchin, Lamberto (1998). "Material and Obsolescence on Flute Tone
Quality."ASA Publications, Vol.103 (5). N.p., 1998. Web. Sept. 2012.
<http://asadl.org/jasa/resource/1/jasman/v103/i5/p2835_s4?bypassSSO=1>.
Healy, Skip. "Feel the Bore." Healy Flute Company, n.d. Web.
<http://www.skiphealy.com/music/tips/24.htm>.
Hopkin, Bart (1996).Musical Instrument Design: Practical Information for Instrument Making.
Tucson, AZ: See Sharp, 1996. Print.
Linortner, R. (2001). Silver, Gold, Platinum- and the Sound of the Flute. Universitat fur Musik
und Darstellende Kunst Wien. Web. Sept. 2012
www.xmarks.com/site/iwk.mdw.ac.at/Forschung/english/linortner/e.htm.
McGee, T (2004). "Flute Myths Exploded!" McGee Flutes. N.p., 11 Mar. 2004. Web. Sept. 2012.
<http://www.mcgee-flutes.com/FluteMyths.htm>.
60
Kennedy, M. (2007). The Concise Oxford Dictionary of Music. New York: Oxford University Press.
Tattersall, Malcolm (2008). "Does It Matter What It's Made Of?"Woodwind Materials. N.p., May
2008. Web. Sept. 2012.
<http://members.iinet.net.au/~mtattersall/Articles/Woodwind_Materials.htm>.
Transverse Flute in F. 1978. Photograph. Metropolitan Museum of Art, New York.Metropolitan
Museum of Art. Metropolitan Museum of Art. Web. 25 Mar. 2013.
<http://www.metmuseum.org/Collections/search-the-
collections/180016501?rpp=60&pg=1&rndkey=20130325&ft=*&what=Wood%7cFlutes&p
os=58#fullscreen>.
Welch, Christopher. “Boehm, Nicholson, and the English Flute. Sept. 2012.
<http://www.standingstones.com/irflute2.html>
Wilcocks, Gerda R (2006). "Improving Tone Production on the Flute with Regards to
Embouchure, Lip Flexibility, Vibrato and Tone Colour, as Seen from a Classical Music
Perspective." University of Pretoria, Dec. 2006. Web.
<http://upetd.up.ac.za/thesis/available/etd-09132007-
163345/unrestricted/dissertation.pdf>.
Xorys. "What Is an Irish Flute? (Part 1)."What Is an Irish Flute? (Pt. 1). N.p., n.d. Web. Sept.
2012. <http://www.standingstones.com/irflute.html>.
61
Appendix A
Figure 46: All required values and equations needed to compute tone hole placement in an Excel spreadsheet
Flute
Nearest
tone
hole
open
Freq
Desired Temp in c
(C)
Speed of
Sound
(Ls)Act.
wave
length
(Lmeas)
Dis. From
emb to
tone
(de) Emb
dia
(dbe)Bor
e emb
dia
(ho)
Chimney
at emb
(dh)
Tone Dia
(dend)
Dia at
end of
flute
(hend)
Chimney
at tone hest dest (dbe-de) (dest-dh) Abe-Ae Aest-Ah delle dellh La (mm)
cal-meas
(mm)
RP D4 End 293.66 25 346000 589.1167 517.8985 11.5 17.17703 5.11047 NA 12.78 5.465 5.677028 #VALUE! 127.8631 #VALUE! 27.63336 41.105 520.3783 2.479765
RP E4 6 329.63 25 346000 524.8309 427.891 11.5 17.17703 5.11047 6.23 14.12134 5.258259 5.421574 13.31859 5.677028 7.088586 127.8631 108.8341 27.63336 62.28156 434.916 7.024909
RP F#4 5 369.99 25 346000 467.5802 386.641 11.5 17.17703 5.11047 10.17 14.67298 5.171828 5.372848 13.9229 5.677028 3.752904 127.8631 71.01441 27.63336 52.95045 386.9964 0.355351
RP G4 4 392 25 346000 441.3265 359.811 11.5 17.17703 5.11047 9.08 15.08573 5.111456 5.346276 14.25246 5.677028 5.172464 127.8631 94.78682 27.63336 53.61819 360.075 0.263942
RP A4 3 440 25 346000 393.1818 305.8177 11.5 17.17703 5.11047 7.4 16.88524 5.138118 5.288327 14.97118 5.677028 7.571177 127.8631 133.0277 27.63336 55.16197 310.3865 4.568745
RP B4 2 493.88 25 346000 350.2875 272.9277 11.5 17.17703 5.11047 9.13 16.37985 5.050998 5.223278 15.77793 5.677028 6.647933 127.8631 130.0511 27.63336 50.90874 271.7454 -1.18232
RP C#5 1 554.37 25 346000 312.0659 232.7527 11.5 17.17703 5.11047 8.04 15.91275 4.960411 5.15025 16.68366 5.677028 8.643661 127.8631 167.8419 27.63336 51.51443 232.9182 0.165422
RP D5 End 587.33 25 346000 589.1066 517.8985 11.5 17.17703 5.11047 NA 12.78 5.465 5.110458 17.17718 5.677028 #VALUE! 127.8631 #VALUE! 27.63336 34.21488 527.2584 9.359852
RP E5 6 659.26 25 346000 524.8309 427.891 11.5 17.17703 5.11047 6.23 14.12134 5.258259 5.023618 18.2542 5.677028 12.0242 127.8631 231.2235 27.63336 54.28163 442.9159 15.02484
RP F#5 5 739.99 25 346000 467.5739 386.641 11.5 17.17703 5.11047 10.17 14.67298 5.171828 4.926154 19.46299 5.677028 9.292986 127.8631 216.2821 27.63336 46.16819 393.7723 7.131295
RP G5 4 783.99 25 346000 441.3322 359.811 11.5 17.17703 5.11047 9.08 15.08573 5.111456 4.873034 20.12181 5.677028 11.04181 127.8631 253.2443 27.63336 47.17142 366.5274 6.716344
RP A5 3 880 25 346000 393.1818 305.8177 11.5 17.17703 5.11047 7.4 16.88524 5.138118 4.757123 21.55938 5.677028 14.15938 127.8631 322.0501 27.63336 49.49576 316.0527 10.23496
RP B5 2 987.77 25 346000 350.284 272.9277 11.5 17.17703 5.11047 9.13 16.37985 5.050998 4.627014 23.17304 5.677028 14.04304 127.8631 356.2826 27.63336 45.72378 276.9268 3.999091
RP C#6 1 1108.73 25 346000 312.0688 232.7527 11.5 17.17703 5.11047 8.04 15.91275 4.960411 4.480982 24.9842 5.677028 16.9442 127.8631 439.4842 27.63336 47.06126 237.3741 4.6214
RP D6 End 1174.66 25 346000 589.1066 517.8985 11.5 17.17703 5.11047 NA 12.78 5.465 4.401386 25.97138 5.677028 #VALUE! 127.8631 #VALUE! 27.63336 30.76994 530.7033 12.80479
62
Figure 47: Mopane Wood D Flute Frequency Test
Figure 48: Olive Wood D Flute Frequency Test
Note Actual Freq Test 1 Test 2 Test3 Average Test 1 Test 2 Test3 Average
D4 293.66 294.6 296 296.4 295.6667 0.94 2.34 2.74 2.007
E4 329.63 333.5 331.2 336.7 333.8 3.87 1.57 7.07 4.170
F#4 369.99 372 370.8 374.5 372.4333 2.01 0.81 4.51 2.443
G4 392 395.7 397.5 398.4 397.2 3.7 5.5 6.4 5.200
A4 440 448 444.5 449 447.1667 8 4.5 9 7.167
B4 493.88 500.8 500.1 504.4 501.7667 6.92 6.22 10.52 7.887
C#5 554.37 558.2 557.3 561.7 559.0667 3.83 2.93 7.33 4.697
D5 587.33 598.1 596.6 599.4 598.0333 10.77 9.27 12.07 10.703
E5 659.26 672.8 672.1 680.3 675.0667 13.54 12.84 21.04 15.807
F#5 739.99 750.5 754.2 762.7 755.8 10.51 14.21 22.71 15.810
G5 783.99 804.2 804.2 806.5 804.9667 20.21 20.21 22.51 20.977
A5 880 905.4 897.2 909.4 904 25.4 17.2 29.4 24.000
B5 987.77 1014 1003 1020 1012.333 26.23 15.23 32.23 24.563
C#6 1108.73 1127 1116 1133 1125.333 18.27 7.27 24.27 16.603
D6 1174.66 1207 1202 1209 1206 32.34 27.34 34.34 31.340
Mopane Wood D Flute Frequency TestFrequency (Hz) Difference between Played Frequency and Actual Frequency (Hz)
Note Actual Freq Test 1 Test 2 Test3 Average Test 1 Test 2 Test3 Average
D4 293.66 293.6 294.1 294.4 294.0333 -0.06 0.44 0.74 0.373
E4 329.63 333.6 336 332.8 334.1333 3.97 6.37 3.17 4.503
F#4 369.99 370.4 369.3 367.5 369.0667 0.41 -0.69 -2.49 -0.923
G4 392 396.5 394.2 393.5 394.7333 4.5 2.2 1.5 2.733
A4 440 447.2 445.2 444.6 445.6667 7.2 5.2 4.6 5.667
B4 493.88 500.9 500.8 501.6 501.1 7.02 6.92 7.72 7.220
C#5 554.37 556.7 551.5 550.3 552.8333 2.33 -2.87 -4.07 -1.537
D5 587.33 598.6 592 594 594.8667 11.27 4.67 6.67 7.537
E5 659.26 677.3 675.7 672.4 675.1333 18.04 16.44 13.14 15.873
F#5 739.99 751.2 754.7 750 751.9667 11.21 14.71 10.01 11.977
G5 783.99 807.4 808.7 805.5 807.2 23.41 24.71 21.51 23.210
A5 880 903.6 908.2 899.7 903.8333 23.6 28.2 19.7 23.833
B5 987.77 1013 1020 992.6 1008.533 25.23 32.23 4.83 20.763
C#6 1108.73 1125 1128 1135.7 1129.567 16.27 19.27 26.97 20.837
D6 1174.66 1200 1209 1192 1200.333 25.34 34.34 17.34 25.673
Frequency (Hz) Difference between Played Frequency and Actual Frequency (Hz)
Olive Wood D Flute Frequency Test
63
Figure 49: Olive Wood Eb Flute Frequency Test
Figure 50: Difference between Predicted Change and Actual Change in Frequency due to Bore Extension
Figure 51: Frequency Data Collected for Chimney Height Increases
Note Actual Freq Test 1 Test 2 Test3 Average Test 1 Test 2 Test3 Average
Eb4 311.13 310.5 310.2 310 310.2333 -0.63 -0.93 -1.13 -0.897
F4 349.23 346.8 347.8 345.4 346.6667 -2.43 -1.43 -3.83 -2.563
G4 392 385.2 385.4 389.5 386.7 -6.8 -6.6 -2.5 -5.300
Ab4 415.3 412.6 409.4 416.1 412.7 -2.7 -5.9 0.8 -2.600
Bb4 466.16 468.5 462.4 463.6 464.8333 2.34 -3.76 -2.56 -1.327
C5 523.25 521.2 515.8 518.3 518.4333 -2.05 -7.45 -4.95 -4.817
D5 587.33 579.2 572.2 577.6 576.3333 -8.13 -15.13 -9.73 -10.997
Eb5 622.25 625.4 627.5 624.2 625.7 3.15 5.25 1.95 3.450
F5 698.46 693.2 697.7 693.7 694.8667 -5.26 -0.76 -4.76 -3.593
G5 783.99 777.6 777.7 779.7 778.3333 -6.39 -6.29 -4.29 -5.657
Ab5 830.61 827.7 828.6 836.9 831.0667 -2.91 -2.01 6.29 0.457
Bb5 932.33 925.6 931.2 942.3 933.0333 -6.73 -1.13 9.97 0.703
C6 1046.5 1040 1054 1055 1049.667 -6.5 7.5 8.5 3.167
D6 1174.66 1162 1172 1170 1168 -12.66 -2.66 -4.66 -6.660
Eb6 1244.51 1255 1261 1268 1261.333 10.49 16.49 23.49 16.823
Frequency (Hz)
Olive Wood Eb Flute Frequency TestDifference between Played Frequency and Actual Frequency (Hz)
3.82 7.03 10.52 3.82 7.03 10.52 3.82 7.03 10.52
D4 0.7 3.7 4.6 D4 2.57935 4.528173 6.646987 D4 1.87935 0.828173 2.046987
E4 2 5.8 7.7 E4 2.931769 5.146862 7.555171 E4 0.931769 -0.65314 -0.14483
F#4 5.2 7.1 8 F#4 3.300248 5.793744 8.504741 F#4 -1.89975 -1.30626 0.504741
G4 5.3 9.6 11.1 G4 3.537573 6.21038 9.116328 G4 -1.76243 -3.38962 -1.98367
A4 5.3 7.3 12.9 A4 3.990811 7.00606 10.28432 A4 -1.30919 -0.29394 -2.61568
B4 5.2 11.7 18.6 B4 9.016573 15.82903 23.23572 B4 3.816573 4.129028 4.635715
C#5 14.5 19.9 25.6 C#5 10.17108 17.85582 26.21088 C#5 -4.32892 -2.04418 0.61088
D5 7 10.3 12.2 D5 5.338033 9.371173 13.75611 D5 -1.66197 -0.92883 1.556114
E5 5.6 14.1 14.7 E5 6.007183 10.5459 15.48051 E5 0.407183 -3.5541 0.780514
F#5 8.1 16.9 18.2 F#5 6.775367 11.89448 17.46012 F#5 -1.32463 -5.00552 -0.73988
G5 9.5 16.7 15.2 G5 7.246448 12.72149 18.6741 G5 -2.25355 -3.97851 3.474102
A5 11 22.3 22.4 A5 8.128834 14.27056 20.94801 A5 -2.87117 -8.02944 -1.45199
B5 10 21.3 22.8 B5 9.10044 15.97626 23.45184 B5 -0.89956 -5.32374 0.65184
C#6 12 27 31 C#6 10.01048 17.57389 25.79702 C#6 -1.98952 -9.42611 -5.20298
D6 15 21 25 D6 10.73317 18.84259 27.65938 D6 -4.26683 -2.15741 2.659376
Distances (mm) Distances (mm)
Dif
fere
nce
of
Pre
dic
ted
Ch
ange
an
d A
ctu
al (
Hz)
Freq
uen
cy D
iffe
ren
ce f
rom
Bas
e (H
z)
Distances (mm)
Pre
dic
ted
Fre
qu
ency
Dif
fere
nce
fro
m B
ase
(Hz)
Frequency
Played
(Hz)
Chimney
Height
(mm)
Frequency
Played
(Hz)
Chimney
Height
(mm)
Frequency
Played
(Hz)
Chimney
Height
(mm)
Frequency
Played
(Hz)
Chimney
Height
(mm)
Frequency
Played
(Hz)
Chimney
Height
(mm)
Frequency
Played
(Hz)
Chimney
Height
(mm)
Frequency
Played
(Hz)
Chimney
Height
(mm)
Frequency
Played
(Hz)
Chimney
Height
(mm)
Frequency
Played
(Hz)
Chimney
Height
(mm)
287.4 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
326.2 0 320.1 0.75 322.8 1.76 318.8 3.11 317.2 3.51 312.3 5.45 313.3 6.09 310.4 7.41 310.3 8.8
368.1 0 363.3 0.96 359.9 2.63 360.6 3.61 359.2 4.53 358.7 5.88 355.7 7.44 352.6 8.2 356.9 9.36
391.4 0 383.7 1.06 385.2 2.03 385.8 3.18 384.8 4.18 384 5.54 378.3 6.49 382.1 7.87 382.5 8.89
446 0 436.1 0.79 434.3 2.13 429.1 2.83 433.5 4.79 430.6 5.68 430.8 6.63 427.2 8.13 425.6 9.22
502.3 0 489.7 0.89 487.6 1.95 477.5 3 479.4 4.21 479.8 5.3 482.3 6.47 482.1 7.8 485.1 9.07
557.3 0 561.8 0.61 538.1 2.25 547.1 3.36 544.2 4.76 536.3 5.14 551.6 6.42 553.6 7.13 549.1 8.98
596.9 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
670.9 0 664.2 0.75 662.4 1.76 665.9 3.11 648.1 3.51 642.5 5.45 638.5 6.09 632.7 7.41 636.7 8.8
754.3 0 747.4 0.96 741 2.63 738.2 3.61 741.2 4.53 727.1 5.88 727.2 7.44 725.3 8.2 730.1 9.36
804.5 0 797.7 1.06 798.6 2.03 784.2 3.18 794.1 4.18 785.3 5.54 776.2 6.49 780.6 7.87 784.1 8.89
908.6 0 884.3 0.79 879.6 2.13 868 2.83 873.1 4.79 864.7 5.68 864.8 6.63 860.7 8.13 895.7 9.22
1025 0 976.6 0.89 978.2 1.95 967.2 3 962.3 4.21 948.6 5.3 962.6 6.47 965.6 7.8 962 9.07
1117 0 1077 0.61 1074 2.25 1091 3.36 1076 4.76 1063 5.14 1083 6.42 1081 7.13 1076 8.98
1207 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
Number of Cards used and Frequencies Played
0 1 3 155 7 9 11 13